How to apply Corrective Smooth as Shapekey?
This is a quirk of how the Apply as Shapekey works. The mechanism is outside the specific modifier's logic. 

Essentially when you click Apply as Shapekey blender takes the unmodified mesh and adds just the modifier that your are applying to the mesh, then uses the resulting shape. For corrective smooth, this means nothing happens, as corrective smooth needs other deformation in order to do anything.

There is however a workaround for this. Blender comes bundled with and addon called Corrective Shape Key which can generate a shape from the whole combined mesh as a corrective shape. To do this you need to first use the addon to create a duplicate for editing, and then disable the corrective smooth, and then add the dupe back as new shape using the addon's functionality to reverse transforms.